Tamiya has more than one King Tiger kit. The one I built doesn't glue the sprocket onto its axle. It uses a little plastic insert that enables the sprocket to turn. If your kit is like that, then you should be able to pull the sprocket off and wrap the track around it before the glue on the links dries completely. You can then push the sprocket back onto the axle with the tracks wrapped around it. It sounds like you may also have glued the fenders on, so that will make it a little harder, but it's still doable. You'll notice that the fenders come down so far on the sides that you can't really see the top run of the tracks, so you don't need to try to make the top run perfect. I read where someone said he didn't even bother putting on the top run of the tracks because it wasn't viewable, so just focus on the links that show and worry about the rest later.
As an alternative, if you used Tamiya glue, you can dissolve it with thinner and separate the fenders and the hull and start over. Just drench the seams with the thinner, and the glue will loosen after a while and you can pull the parts apart.
I hope this helps. You'll like the look with the individual links a whole lot better than the band it came with.